### Video Transcript

Given that ๐ด is the three-by-three
matrix three, zero, negative two, negative two, one, ๐ฅ, zero, three, seven, find
the value of ๐ฅ such that ๐ด squared is equal to the three-by-three matrix nine,
negative six, negative 20, negative eight, negative 20, negative 52, negative six,
24, 28.

In this question, we are given a
three-by-three matrix ๐ด containing an unknown value of ๐ฅ. We need to find this value of
๐ฅ. To do this, we are given the matrix
๐ด squared. We can recall that we define
squaring a matrix in the same way we square a number. We multiply it by itself, so ๐ด
squared is ๐ด times ๐ด. We can use this to find ๐ด
squared. We need to multiply ๐ด by
itself. This gives us the following product
for ๐ด squared. Before we calculate the square of
this matrix, we can replace ๐ด squared in the equation with the given matrix ๐ด
squared.

We now recall that we multiply
matrices by finding the sum of the products of each entry in each row of the first
matrix with the corresponding elements in the columns of the second matrix. For instance, we can apply this to
the second row of the first matrix and the third column of the second matrix to get
the element in the second row and third column of ๐ด squared. Equating the entry in the second
row and third column of ๐ด squared with the sum of the products of these entries
gives us the following linear equation in ๐ฅ.

We can then solve this equation for
๐ฅ. We simplify each side of the
equation. We then subtract four from both
sides of the equation to obtain negative 56 equals eight ๐ฅ. Then, we divide through by eight to
get that ๐ฅ equals negative seven.